A while ago Joel Hamkins suggested that I copy cat this post of Norman’s: Chart of large cardinals near high-jump with my own large cardinal chart. So here is the large cardinal chart of Ramsey-like cardinals and their relationship to other large cardinal notions. The Ramsey-like cardinals: $\alpha$-*iterable* cardinals ($1\leq\alpha\leq\omega_1$), *strongly Ramsey* cardinals, *super Ramsey* cardinals, and *superlatively Ramsey* cardinals were introduced by myself and/or Philip Welch. All relevant definitions are here.

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- Comment by Victoria Gitman on Forcing in GBC, the ctm approachWhat if we define that a "finite" fragment of GBC asserts the existence of all $\Sigma_n$-definable (with class parameters) classes for some fixed $n$? You can certainly get set models of this theory and you should be able to argue that for any finite collection of sentences forced by $\mathbb P$ this much of the […]Victoria Gitman
- Comment by Victoria Gitman on Taller models of ZFC@Mirco: Maybe (2) should be modified to say that $M=V_{\eta_0}^N$. It seems this would capture better the notion of "making a model taller".Victoria Gitman
- Comment by Victoria Gitman on Is there any elementary embedding characterization for $\Pi_{1}^{1}$ - reflecting cardinals?It sounds like you are talking about a $\Pi^1_1$-indescribable cardinal. I think a cardinal $\kappa$ is $\Pi^1_1$-reflecting if it is inaccessible and $V_\kappa\prec_{\Pi_1} V$.Victoria Gitman

- Comment by Victoria Gitman on Forcing in GBC, the ctm approach
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Yay! I love it!